Let the given statement
T(n) = 102n–1+1 be a multiple of 11
⇒ 102n–1+1 = M(11)
Basic Step:
For n = 1, 102×1–1+1 = 10+1 = 11 which is divisible by 11.
Induction Step:
Assume that T(k) = 102k–1+1 is divisible by 11.
⇒ 102k–1+1 = M(11) ∀ n∈N .....(i)
Then, we now show that T(k + 1) is true.
T(k + 1) = 102(k +1)–1+1 = 102k–1+2 + 1 = 102 . 102k–1+1
= 100 (M (11) – 1) + 1 (From (i))
= 100 . M(11) – 100 + 1 = 100 . M (11) – 99
⇒ T(k + 1) is divisible by 11, when T(k) is divisible by 11.
⇒ T(n) holds true for all n∈N.